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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2016
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1606.02449 |
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| _version_ | 1866912789416640512 |
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| author | Benjamini, Itai Tessera, Romain |
| author_facet | Benjamini, Itai Tessera, Romain |
| contents | Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg is whether there exists a two-sided infinite geodesic in first passage percolation on Z^2, and more generally on Z^n for n>1. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite exponential moment for the random lengths, we prove that if a graph X has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost surely, there exists a bi-infinite geodesic in first passage percolation on X. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1606_02449 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | First Passage percolation on a hyperbolic graph admits bi-infinite geodesics Benjamini, Itai Tessera, Romain Probability 82B43, 51F99, 97K50 Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg is whether there exists a two-sided infinite geodesic in first passage percolation on Z^2, and more generally on Z^n for n>1. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite exponential moment for the random lengths, we prove that if a graph X has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost surely, there exists a bi-infinite geodesic in first passage percolation on X. |
| title | First Passage percolation on a hyperbolic graph admits bi-infinite geodesics |
| topic | Probability 82B43, 51F99, 97K50 |
| url | https://arxiv.org/abs/1606.02449 |